Groups having complete bipartite divisor graphs for their conjugacy class sizes

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G \ Z(G) (where Z(G) denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p, n} being an edge if gcd(p, n) ≠ 1. In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K<inf>2,5</inf>, giving a solution to a question of Taeri [15].